Janko group

In the area of modern algebra known as group theory, the Janko groups are the four sporadic simple groups J₁, J₂, J₃ and J₄ introduced by Zvonimir Janko. Unlike the Mathieu groups, Conway groups, or Fischer groups, the Janko groups do not form a series, and the relation among the four groups is mainly historical rather than mathematical.

History

Janko constructed the first of these groups, J₁, in 1965 and predicted the existence of J₂ and J₃. In 1976, he suggested the existence of J₄. Later, J₂, J₃ and J₄ were all shown to exist.

J₁ was the first sporadic simple group discovered in nearly a century: until then only the Mathieu groups were known, M₁₁ and M₁₂ having been found in 1861, and M₂₂, M₂₃ and M₂₄ in 1873. The discovery of J₁ caused a great "sensation"[1] and "surprise"[2] among group theory specialists. This began the modern theory of sporadic groups.

And in a sense, J₄ ended it. It would be the last sporadic group (and, since the non-sporadic families had already been found, the last finite simple group) predicted and discovered, though this could only be said in hindsight when the Classification theorem was completed.

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References

Dieter Held, Die Klassifikation der endlichen einfachen Gruppen Archived 2013-06-26 at the Wayback Machine (the classification of the finite simple groups), Forschungsmagazin der Johannes Gutenberg-Universität Mainz 1/86
The group theorist Bertram Huppert said of J₁: "There were a very few things that surprised me in my life... There were only the following two events that really surprised me: the discovery of the first Janko group and the fall of the Berlin Wall."

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[1] Dieter Held, Die Klassifikation der endlichen einfachen Gruppen Archived 2013-06-26 at the Wayback Machine (the classification of the finite simple groups), Forschungsmagazin der Johannes Gutenberg-Universität Mainz 1/86

[2] The group theorist Bertram Huppert said of J₁: "There were a very few things that surprised me in my life... There were only the following two events that really surprised me: the discovery of the first Janko group and the fall of the Berlin Wall."